1. Field of the Invention
The present invention relates generally to the field of measuring attributes of a physical system, and more particularly to methods of measuring a non-symmetric physical function.
2. Description of the Related Art
Various optical devices are based on induced second-order susceptibilities in silica-based glass waveguides (e.g., electro-optic modulators, switches, parametric amplifiers). For example, G. Bonfrate et al. describe optical parametric oscillators useful for the study of quantum interference and quantum cryptography, and for metrology applications in Parametric Fluorescence in Periodically Poled Silica Fibers, Applied Physics Letters, Vol. 75, No. 16, 1999, pages 2356–2358, which is incorporated in its entirety by reference herein. Second-order susceptibility can be induced in a several-microns-thick region of fused silica (a material that is not normally centro-symmetric, and therefore normally does not exhibit a second-order susceptibility) by poling at elevated temperatures. This phenomenon has been described by R. A. Myers et al. in Large Second-Order Nonlinearity in Poled Fused Silica, Optics Letters, Vol. 16, No. 22, 1991, pages 1732–1734, which is incorporated in its entirety by reference herein.
FIGS. 1A and 1B schematically illustrate the poling of a silica wafer 1. As schematically illustrated in FIG. 1A, poling typically comprises using an anode electrode 2 placed in proximity to one surface 3 of the wafer 1 and a cathode electrode 4 placed in proximity to the opposite surface 5 of the wafer 1. A voltage is applied across the wafer 1 for a period of time, resulting in a second-order optical nonlinearity profile having a thickness and localized beneath the surface 3 where the anode electrode was placed, as schematically illustrated in FIG. 1B. Such a poling procedure is described in more detail by Thomas G. Alley et al. in Space Charge Dynamics in Thermally Poled Fused Silica, Journal of Non-Crystalline Solids, Vol. 242, 1998, pages 165–176, which is incorporated herein in its entirety.
The field of poled silica has suffered from the lack of a common method to reliably measure the second-order optical nonlinearity profile of poled samples. This absence of a reliable procedure for measuring nonlinearity profiles may be the reason, at least in part, for wide discrepancies in the measured magnitudes and the positions of the nonlinearity profiles of various poled systems as reported in the literature. The Maker fringe (MF) technique is the most common method currently used to investigate the nonlinearity profile of poled silica. The MF technique comprises focusing a pulsed laser beam of intensity I1 (known as the fundamental signal) onto a sample at an incident angle θ and measuring the intensity I2 of the second harmonic (SH) signal generated within the nonlinear region as a function of the incident angle θ. For a transverse magnetic (TM) polarized fundamental laser beam, the conversion efficiency ηTM(θ) is given by:
                                          η            TM                    ⁡                      (            θ            )                          =                                            I              2                                      I              1                                =                                    f              ⁡                              (                                  θ                  ,                                      n                    1                                    ,                                      n                    2                                                  )                                      ⁢                                                                            ∫                                                                                    ⅆ                        33                                            ⁢                                              (                        z                        )                                                              ⁢                                          ⅇ                                              jΔ                        ⁢                                                                                                  ⁢                                                  k                          ⁡                                                      (                            θ                            )                                                                          ⁢                        z                                                              ⁢                                          ⅆ                      z                                                                                                  2                                                          (        1        )            where    d33(z) is the nonlinear coefficient (which is proportional to the second-order susceptibility χ(2));    z is the direction normal to the sample surface (i.e., parallel to the poling field);    n1 and n2 are the refractive indices at the fundamental and SH frequencies, respectively;    Δk=k2−2k1, where k1 and k2 are the fundamental and SH wave numbers, respectively, and    ƒ(θ, n1, n2) is a well-defined function of the incident angle θ (relative to the surface normal direction) and refractive indices n1 and n2.
The function ƒ(θ, n1, n2) accounts for both the power loss due to reflection suffered by the fundamental and the SH beams, and the projection of the input electric field along the appropriate direction. In general, ƒ(θ, n1, n2) depends on both the polarization of the input fundamental wave and the geometry of the second harmonic generation configuration. The exact formula of ƒ(θ, n1, n2) is given by D. Pureur, et al. in Absolute Measurement of the Second-Order Nonlinearity Profile in Poled Silica, Optics Letters, Vol. 23, 1998, pages 588–590, which is incorporated in its entirety by reference herein. This phenomenon is also described by P. D. Maker et al. in Effects of Dispersion and Focusing on the Production of Optical Harmonics, Physics Review Letters, Vol. 8, No. 1, 1962, pages 21–22, which is incorporated in its entirety by reference herein.
The conversion efficiency ηTM(θ) is obtained experimentally by rotating the sample with respect to the incident laser beam and measuring the power of the SH signal as a function of the incident angle θ. Due to dispersion of the laser beam, Δk is finite and ηTM(θ) exhibits oscillations (called the Maker fringes) which pass through several maxima and minima. The objective of this measurement is to retrieve the second-order nonlinearity profile d33(z). The absolute value of the integral in Equation 1 is the amplitude of the Fourier transform of d33(z). In principle, if one knew both the amplitude and the phase of a Fourier transform, one can readily infer the argument of the Fourier transform (in this case d33(z)) by taking the inverse Fourier transform of the Fourier transform. However, the measured Maker fringes provide only the magnitude of the Fourier transform, not its phase. Consequently, for an arbitrary and unknown nonlinearity profile, the MF measurement alone is not sufficient to determine a unique solution for d33(z). Even if the phase information were available, it would provide the shape of d33(z) but it would still not provide the location of this shape beneath the surface of the sample (i.e., where the nonlinearity profile starts beneath the surface).
Previous efforts to determine d33(z) have involved fitting various trial profiles to the measured MF data. Examples of such efforts are described by M. Qiu et al. in Double Fitting of Maker Fringes to Characterize Near-Surface and Bulk Second-Order Nonlinearities in Poled Silica, Applied Physics Letters, Vol. 76, No. 23, 2000, pages 3346–3348; Y. Quiquempois et al. in Localisation of the Induced Second-Order Non-Linearity Within Infrasil and Suprasil Thermally Poled Glasses, Optics Communications, Vol. 176, 2000, pages 479–487; and D. Faccio et al. in Dynamics of the Second-Order Nonlinearity in Thermally Poled Silica Glass, Applied Physics Letters, Vol. 79, No. 17, 2001, pages 2687–2689. These references are incorporated in their entirety by reference herein.
However, the previous methods do not produce a unique solution for d33(z). Two rather different trial profiles can provide almost equally good fits to the measured MF data. This aspect of using fitting routines to determine d33(z) is described in more detail by Alice C. Liu et al. in Advances in the Measurement of the Poled Silica Nonlinear Profile, SPIE Conference on Doped Fiber Devices II, Boston, Mass., November 1998, pages 115–119, which is incorporated in its entirety by reference herein.